We establish the following results on higher order
${\mathcal{S}}^{p}$
-differentiability,
$1<p<\infty$
, of the operator function arising from a continuous scalar function
$f$
and self-adjoint operators defined on a fixed separable Hilbert space:
- (i)
$f$
is
$n$
times continuously Fréchet
${\mathcal{S}}^{p}$
-differentiable at every bounded self-adjoint operator if and only if
$f\in C^{n}(\mathbb{R})$
;
- (ii)if
$f^{\prime },\ldots ,f^{(n-1)}\in C_{b}(\mathbb{R})$
and
$f^{(n)}\in C_{0}(\mathbb{R})$
, then
$f$
is
$n$
times continuously Fréchet
${\mathcal{S}}^{p}$
-differentiable at every self-adjoint operator;
- (iii)if
$f^{\prime },\ldots ,f^{(n)}\in C_{b}(\mathbb{R})$
, then
$f$
is
$n-1$
times continuously Fréchet
${\mathcal{S}}^{p}$
-differentiable and
$n$
times Gâteaux
${\mathcal{S}}^{p}$
-differentiable at every self-adjoint operator.
We also prove that if
$f\in B_{\infty 1}^{n}(\mathbb{R})\cap B_{\infty 1}^{1}(\mathbb{R})$
, then
$f$
is
$n$
times continuously Fréchet
${\mathcal{S}}^{q}$
-differentiable,
$1\leqslant q<\infty$
, at every self-adjoint operator. These results generalize and extend analogous results of Kissin et al. (Proc. Lond. Math. Soc. (3)108(3) (2014), 327–349) to arbitrary
$n$
and unbounded operators as well as substantially extend the results of Azamov et al. (Canad. J. Math.61(2) (2009), 241–263); Coine et al. (J. Funct. Anal.; doi:10.1016/j.jfa.2018.09.005); Peller (J. Funct. Anal.233(2) (2006), 515–544) on higher order
${\mathcal{S}}^{p}$
-differentiability of
$f$
in a certain Wiener class, Gâteaux
${\mathcal{S}}^{2}$
-differentiability of
$f\in C^{n}(\mathbb{R})$
with
$f^{\prime },\ldots ,f^{(n)}\in C_{b}(\mathbb{R})$
, and Gâteaux
${\mathcal{S}}^{q}$
-differentiability of
$f$
in the intersection of the Besov classes
$B_{\infty 1}^{n}(\mathbb{R})\cap B_{\infty 1}^{1}(\mathbb{R})$
. As an application, we extend
${\mathcal{S}}^{p}$
-estimates for operator Taylor remainders to a broad set of symbols. Finally, we establish explicit formulas for Fréchet differentials and Gâteaux derivatives.